Abbas' embattled party ran virtually unopposed, and Saturday's election was seen as a litmus test of Fatah's popularity.
Underlining the political schism, about 800,000 Palestinians were expected to vote for representatives in 145 local councils in the West Bank, but not in the Gaza Strip.
"These are relatively minor council elections", Fawcett added.
Several months ago, Hamas announced that it had elected senior military official Yehya Senwar as its leader in the Gaza Strip.
Hamas took part in the first local elections held in 2005, which chose nearly 100 local councils, as the process was never completed, then in parliamentary elections in 2006 marking a sweeping victory by winning 74 seats out of the 132 Legislative Council elections.
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In Nablus, the biggest West Bank city, turnout was only 20 percent; in Al Bireh, Ramallah's twin city, 28 percent; and Hebron, the second-largest West Bank city, 30 percent.
The Palestinian president is now more than a decade into what was supposed to be a four-year term.
Speaking to reporters after the vote, Hamdallah said the elections "send a message to the world that our people deserve a state as any other nation in the world".
Hamas said The Palestinian Authority had made a unilateral decision to go ahead with the vote before an agreement on a legal framework had been reached.
In just two cities, Jenin and Jericho, did Fatah win an outright victory without support from other participating factions.
"The result wasn't great for us", said Tayseer Abu Sneineh, the head of Fatah's list in the city.
"This is the traditional elections tactic of reaching out to tribes", said Khaled Qawasmi, a former Fatah official who was sacked for running independently.
Palestinians typically vote in municipal elections according to political bases.
Electoral worker gives a ballot to a voter at a polling station in the West Bank city of Nablus, Saturday, May 13, 2017.
Official figures for Saturday's election showed turnout at 53.4 per cent - almost the same as the turnout for local elections in the West Bank in 2012, said electoral commission chief Hanna Nasser in Ramallah.